Find each of the following product:

To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., $a^{m} \times a^{n}=a^{m+n}$.

We have:

$(-5 a) \times\left(-10 a^{2}\right) \times\left(-2 a^{3}\right)$

$=\{(-5) \times(-10) \times(-2)\} \times\left(a \times a^{2} \times a^{3}\right)$

$=\{(-5) \times(-10) \times(-2)\} \times\left(a^{1+2+3}\right)$

$=\{(-5) \times(-10) \times(-2)\} \times\left(a^{1+2+3}\right)$

$=-100 a^{6}$

Thus, the answer is $-100 a^{6}$.